OFFSET
0,8
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 3, 8, 15, 24, 35, 48, ...
0, 7, 32, 81, 160, 275, 432, ...
0, 19, 136, 459, 1120, 2275, 4104, ...
0, 51, 592, 2673, 8064, 19375, 40176, ...
0, 141, 2624, 15849, 59136, 168125, 400896, ...
0, 393, 11776, 95175, 439296, 1478125, 4053888, ...
MATHEMATICA
A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, _] = 1; A[_, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, May 05 2019
STATUS
approved